Question

write the function below in the form ( y = f(u) ) and ( u = g(x) ), then find ( \frac{dy}{dx} ) as a function of ( x ).

( y=\tan (cot x) )

what are the functions ( f(u) ) and ( g(x) )?

( f(u)=)
( g(x)=)

Explanation:

Step1: Identify \( f(u) \) and \( g(x) \)

Given \( y=\tan(\cot x) \), we can let \( u = \cot x\) (so \(g(x)=\cot x\)) and \( y = f(u)=\tan u\).

Step2: Find the derivatives of \( f(u) \) and \( g(x) \)

The derivative of \( f(u)=\tan u \) with respect to \( u \) is \( f^{\prime}(u)=\sec^{2}u \).
The derivative of \( g(x)=\cot x\) with respect to \( x \) is \( g^{\prime}(x)=-\csc^{2}x \).

Step3: Apply the chain - rule \( \frac{dy}{dx}=f^{\prime}(u)\cdot g^{\prime}(x) \)

Substitute \( u = \cot x\) into \( f^{\prime}(u) \), we get \( f^{\prime}(u)=\sec^{2}(\cot x) \).
Then \( \frac{dy}{dx}=\sec^{2}(\cot x)\cdot(-\csc^{2}x)=-\sec^{2}(\cot x)\csc^{2}x \).

Answer:

\( f(u)=\tan u\), \(g(x)=\cot x\), \(\frac{dy}{dx}=-\sec^{2}(\cot x)\csc^{2}x\)